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Stream: practice: communication

Topic: CT jargon

Joe Moeller (Aug 29 2023 at 17:11):

Based on earlier parts of the other thread #practice: communication > CT representation online, I'm interested in discussing the relationship CT has with jargon. In particular, I think that by virtue of CT being a "metamathematics", it interacts directly with many if not most other branches of mathematics, and in this interaction it picks up terms and adopts them as general concepts. The first examples that come to mind are limit, Hopf algebra, and groupoid. I'd like to hear other people's examples that fit this trend. But now that I'm thinking about it, it might actually be something like most terms in CT which started in other fields?

Steve Huntsman (Aug 29 2023 at 17:14):

Adjoint

John Baez (Aug 30 2023 at 08:00):

Some categorical terms that started out in other fields:

product
sum
abelian (as in "abelian category")
isomorphism

where in the last item I'll argue that the general categorical concept of isomorphism was abstracted from the word used in group theory, ring theory etc.

(I don't know when and why people first started using the term "isomorphism". Noether's isomorphism theorems???)

John Baez (Aug 30 2023 at 08:08):

retract
section

These general concepts were abstracted from topology.

Damiano Mazza (Aug 30 2023 at 09:58):

Another good example is "presheaf", which newcomers may find a somewhat mysterious way of calling a contravariant functor into $\mathrm{Set}$ (I remember this precise point was the subject of another thread a while ago). Historically, sheaves in topology were only of abelian groups, if I am not mistaken Grothendieck was the first to stress the importance of (pre)sheaves of bare sets, and this abstraction was possible thanks to category theory (and it was, at the same time, a crucial ingredient of the development of category theory itself: a big part of CT was developed to develop topos theory, which started as the study of categories of sheaves of sets).

John Baez (Aug 30 2023 at 10:20):

Yes, that's a good one! There is also some interesting history about how "sheaf" first meant what we now often call an "etale space", while what we now call a "sheaf" on a space was called a "canonical" or "complete" presheaf:

• Michael Weiss, History of terminology: sheaves, presheaves, etc..

David Michael Roberts (Aug 30 2023 at 11:05):

Even in the 1930s people used the word "isomorphism" (eg of groups, but also of vector spaces, modules) to mean "embedding", much like an isometry is not actually an isomorphism in the community of eg Banach or Hilbert spaces. You have people distinguishing between an isomorphism, and an isomorphism onto (which is what we would call an isomorphism). see eg https://mathoverflow.net/a/280271/4177

Bourbaki helped fix the modern definition, of course.

Going backwards in time, you get people talking about "meriedric isomorphism" and "holoedric isomorphism", even in the 1920s, it seems see eg https://mathoverflow.net/questions/290471/who-first-used-the-word-homomorphism#comment720247_290473

David Michael Roberts (Aug 30 2023 at 11:14):

Sophus Lie used the phrase "holoedrisch isomorph" in his work Theorie der Transformationsgruppen , for instance.

Camille Jordan wrote to the effect of that a group $\Gamma$ is said to be isomorphic to another group $G$ if, in modern parlance, there is a surjective homomorphism from $G\to\Gamma$. "Meriedric" meant the isomorphism is not injective, and I think "holoedric" if it is injective.

David Michael Roberts (Aug 30 2023 at 11:16):

Fun discussion in the article https://dipot.ulb.ac.be/dspace/bitstream/2013/127958/3/Structure.pdf, which is where I got the last two paragraphs of info.

John Baez (Aug 30 2023 at 12:33):

Yikes! The next time someone mentions an isomorphism I'll ask if it's "meriedric". And if they're smart they'll wonder if I'm a time traveler from the 1920s.

Damiano Mazza (Aug 30 2023 at 13:12):

The examples so far are all of words having a precise technical definition. Maybe I am stretching the meaning of "jargon" but I think that recurring words/phrases with an informal but sort-of-standardized meaning should also qualify. In a sense, they are even more insidious to a beginner/outsider. For category theory, an example I am thinking of is "yoga". I've heard it several times from category theorists, but never in any other field, so I would say that it is part of CT jargon, although maybe it's not common enough... (I wonder who was the first to use it. Grothendieck? It is certainly in Récoltes and Semailles. So maybe the use began in algebraic geometry and then stuck in category theory).

Damiano Mazza (Aug 30 2023 at 13:15):

Ha, actually, I'm wrong. I just found out that, apparently, the word "yoga" is used in all of mathematics, not just CT...

Todd Trimble (Aug 30 2023 at 13:27):

I like that word "yoga". The way I think of it, "a yoga" is to "informal" as "a calculus" is to "formal" -- usually sort of a semi-formal collective of techniques wielded by experts. But I digress...

Damiano Mazza (Aug 30 2023 at 16:17):

Yes it's a convenient word!

Damiano Mazza (Aug 30 2023 at 16:19):

Another common expression not exactly corresponding to a technical definition is "microcosm principle". I think it's widespread and well-known enough in the community to be considered part of the CT jargon. I'm sure there are more...

Evan Patterson (Aug 30 2023 at 18:12):

I would say that the microcosm principle is a good example of a yoga! Or rather it is part of the yoga of categorification.

Mike Shulman (Aug 30 2023 at 20:32):

"Chasing a diagram"?

Matteo Capucci (he/him) (Aug 31 2023 at 09:26):

I propose: evilness, naturality in a technical sense, universality in a technical say

Tom Hirschowitz (Sep 02 2023 at 19:41):

How about "distributive law"?

John Baez (Sep 04 2023 at 14:27):

Another one is "exact", as in phrases like "left exact functor" to mean a functor that preserves finite limits, and "right exact functor" to mean one that preserves finite colimits.

Mike Shulman (Sep 04 2023 at 19:48):

... and [[exact square]], and (Barr) [[exact category]], and [[Quillen exact category]]...

Mike Shulman (Sep 04 2023 at 19:49):

Which reminds me of the many meanings of "cartesian"...

Josselin Poiret (Sep 05 2023 at 07:48):

Mike Shulman said:

Which reminds me of the many meanings of "cartesian"...

one of my pet peeves, really. Descartes certainly had some good ideas, but he'll be fine if we avoid giving his name to yet one more thing

Patrick Nicodemus (Sep 09 2023 at 00:00):

Apparently you can just use it anywhere there are limits involved. You have Cartesian products, but if a category is Cartesian it has products and equalizers. If a functor is Cartesian, it preserves all finite limits, unless we're talking about monads, in which case it only preserves pullbacks but not the terminal object. A square is Cartesian if it's a pullback square, I guess on the flimsy reasoning that pullbacks are products in the slice category, but Cartesian morphisms in a fibration are then even another step away from that in generality.

Patrick Nicodemus (Sep 09 2023 at 00:17):

I was frustrated by the name $\mathbf{BG}$ for a long time. There's:

• a simplicial set constructed by Eilenberg and Mac Lane for a discrete group $G$
• any space, or perhaps more specifically a CW complex, up to homotopy, which has $\pi_1(X)=G$ and all higher homotopy groups trivial
• the classifying space of a topological group, which classifies principal bundles over a space up to homotopy (Wait these are topological groups now? Surely the Eilenberg-Mac Lane construction isn't what we're talking about here, how are they connected?)
• the "delooping" - https://ncatlab.org/nlab/show/delooping - (uh oh, we're two paragraphs in and hitting stable infinity categories. Danger zone!)
• the category with one object associated to a group or monoid (because categories are spaces because the nerve functor is fully faithful- get on board!)
• the bicategory associated to a monoidal category by regarding it as a one object category (by vague, handwavy generalization of the previous example)
• the simplicial object given by repeatedly tensoring a monoid with itself - this no longer has to be a set theoretic monoid, it could be a monoid in a monoidal category) this comes in two variants, $B(M,M,1)$ and $B(1,M,1)$. The first one is what is usually called $EM$
• an analogous construction on differentially graded associative algebras, see Loday and Vallette

A pet peeve of mine is that, riffing off this article -
https://ncatlab.org/nlab/show/concept+with+an+attitude
Some names like $BG$ do this, but instead of just bringing with them one perspective, they bring with them many perspectives, because the name itself is an attempt to link together many different concepts from different areas of mathematics, understood in totally different ways by the people in those fields. That is, the "concept with attitude" is meant to denote a whole family of perspectives together. And this is really intimidating to someone trying to dive into an article that uses the name unless the author clearly specifies what perspective they want you to have when they are tossing that name around. When a concept has so much baggage, if you leave it open ended to the reader what of that baggage they're meant to be carrying around while they listen to you talk, this is I think very discouraging as there's a lot of fear that they're missing crucial parts of what you're saying that are implicitly contained in this "concept-with-attitude" way of communicating

Patrick Nicodemus (Sep 09 2023 at 00:20):

I would say that if you're considering reusing a heavily-worn name, you have to make it clear to the reader what background is expected of them, because if you don't then you're leaving open the interpretation "every other concept which has ever used the same name is directly relevant to what I'm doing right now."

John Baez (Sep 09 2023 at 09:10):

Patrick Nicodemus said:

I was frustrated by the name $\mathbf{BG}$ for a long time. There's:

1. a simplicial set constructed by Eilenberg and Mac Lane for a discrete group $G$
2. any space, or perhaps more specifically a CW complex, up to homotopy, which has $\pi_1(X)=G$ and all higher homotopy groups trivial
3. the classifying space of a topological group, which classifies principal bundles over a space up to homotopy (Wait these are topological groups now? Surely the Eilenberg-Mac Lane construction isn't what we're talking about here, how are they connected?)

The good news is that these three are "the same" for a discrete group $G$.

More precisely, taking the geometric realization of the simplicial set in 1 is a way to build a CW complex with property 2, and all CW complexes with property 2 are homotopy equivalent. Furthermore, this CW complex is also the classifying space discussed in 3 - where by "is" we mean "is homotopy equivalent to", since any CW complex homotopy equivalent to a classifying space for $G$ is also a classifying space for $G$.

John Baez (Sep 09 2023 at 09:17):

I think your parenthetical remark about the Eilenberg-Mac Lane construction is backwards. When $G$ is discrete people often call $BG$ the Eilenberg-Mac Lane space of $G$ and denote it by $K(G,1)$, since... well... that's what it is, and Eilenberg and Mac Lane dealt with this case. But when $G$ is a non-discrete topological group, $BG$ is a generalization of the Eilenberg-Mac Lane space: unlike an Eilenberg-Mac Lane space, it can have nontrivial $\pi_n$ for $n > 1$. So you're not allowed to call it an Eilenberg-Mac Lane space or denote it by $K(G,1)$.

John Baez (Sep 09 2023 at 09:18):

Seeing why 1-3 are all "the same" in the senses I explained is not trivial but every homotopy theorists should know it.

James Deikun (Sep 09 2023 at 09:28):

Do you mean "denote it by $K(G,1)$" at the end?

John Baez (Sep 09 2023 at 09:29):

Yes, sorry, I'll fix that.

Paolo Perrone (Sep 09 2023 at 10:40):

I think the problem with notation like $BG$ is that many concepts are intimately related, or even the same in some way, so that the notation is well-defined and justified. However, when learning about those concepts, one still doesn't see the equivalence or analogy, and so the notation gets very confusing. (For example, one has sentences like "let's prove that $BG\cong BG$".)

Patrick Nicodemus (Sep 09 2023 at 14:48):

John Baez said:

I think your parenthetical remark about the Eilenberg-Mac Lane construction is backwards. When $G$ is discrete people often call $BG$ the Eilenberg-Mac Lane space of $G$ and denote it by $K(G,1)$, since... well... that's what it is, and Eilenberg and Mac Lane dealt with this case. But when $G$ is a non-discrete topological group, $BG$ is a generalization of the Eilenberg-Mac Lane space: unlike an Eilenberg-Mac Lane space, it can have nontrivial $\pi_n$ for $n > 1$. So you're not allowed to call it an Eilenberg-Mac Lane space or denote it by $K(G,1)$.

Yeah I understand this now. These parentheticals are just expressing confusions I had a few years ago when I first started learning this stuff. I should point out that many textbooks on algebraic topology focus narrowly on one interpretation for the purpose of churning through the material they need to cover, they cut out everything else for efficiency. So your comment "every homotopy theorist should know that these are equivalent" doesn't apply imo to somebody who has read a textbook chapter on BG. You have to pick it up in other ways

Mike Shulman (Sep 09 2023 at 20:11):

Patrick Nicodemus said:

if a category is Cartesian it has products and equalizers.

AFAIK the only source that uses the word that way is Sketches of an Elephant. I think it's more common for a "cartesian category" to mean a category with finite products.

John Baez (Sep 09 2023 at 20:37):

Yes, please let's stomp out Johnstone's definition.

John Baez (Sep 09 2023 at 20:40):

It would be nice to have a quick word for a category with finite limits, but the only phrase I know of people using (except Johnstone's "cartesian") is "finite limits theory", which is a name with attitude. For a functor that preserves finite limits we have a quick name: "left exact", or if you're in a real hurry "lex".

Mike Shulman (Sep 09 2023 at 20:40):

People say "lex category" too.

John Baez (Sep 09 2023 at 20:40):

Okay, good. I was thinking that would be nice.

Mike Shulman (Sep 09 2023 at 20:41):

I can't say I really like the sound of it, but at least it's short and unambiguous.

John Baez (Sep 09 2023 at 20:49):

Patrick Nicodemus said:

So your comment "every homotopy theorist should know that these are equivalent" doesn't apply imo to somebody who has read a textbook chapter on BG. You have to pick it up in other ways.

Right, by a "homotopy theorist" I meant someone who seriously studies the subject. You're right that individual textbook treatments are too focused on getting particular jobs done to convey the overall picture. I tried to efficiently convey some of the big picture here:

• Classifying spaces made easy.

but this is sort of hasty and amateurish.

Matteo Capucci (he/him) (Sep 21 2023 at 13:47):

Josselin Poiret said:

Mike Shulman said:

Which reminds me of the many meanings of "cartesian"...

one of my pet peeves, really. Descartes certainly had some good ideas, but he'll be fine if we avoid giving his name to yet one more thing

proposal to start naming things after Gauss and Euler instead

Todd Trimble (Sep 21 2023 at 14:55):

The only semi-convincing thing I ever heard about applying "cartesian" to describing things in finite limit logic is that Descartes had not only the big idea of cartesian coordinates and product spaces, uniting algebra and geometry, but the idea of geometric figures being loci of equations, and these loci are precisely equalizers of maps between product spaces. I think I first heard this idea from Freyd (is it in Categories, Allegories?).

fosco (Sep 25 2023 at 19:25):

Josselin Poiret said:

Mike Shulman said:

Which reminds me of the many meanings of "cartesian"...

one of my pet peeves, really. Descartes certainly had some good ideas, but he'll be fine if we avoid giving his name to yet one more thing

Joyal suggested to call what we now call a finite limit theory a "Cartesian theory", in that the theory of finite limits may be said to date back to Descartes, who introduced the concepts of a product $\mathbb R \times \mathbb R$ and of an equalizer $\{(x,y)\mid f(x,y) = g(x,y)\}$.

fosco (Sep 25 2023 at 19:28):

ah, @Todd Trimble we had the same remark in mind :heart: I didn't see yours.

fosco (Sep 25 2023 at 19:29):

(by the way my comment was a quote, iirc it's Kock's SDG book.)